Stocking’s Reciprocation :: “Inverting the dividend of a division problem and multiplying by the divisor yields the reciprocal of inverting the divisor of a division problem and multiplying by the dividend.”
Oh no, here he goes again, off on one of his wild math/PEMDAS/Order of Operations tangents. But you know, I’m really not doing anything different in this area than I would do in exegeting a biblical passage. I’m looking for patterns in language and logic, grammar and syntax, that reveals clues to the meaning and intent of the words or figures on a page. I’m applying a scientific method to all this as well, forming new or related hypotheses and testing them out to see whether they can be more than just a harebrained rambling.
That is what’s been happening to me with this whole PEMDAS/Order of Operations obsession I’ve found myself pursuing. Every time I think I’ve got the definitive solution, proof, or argument, I get whacked upside the head by another even more convincing argument. Initially those were coming fast and furious, and I couldn’t keep up with myself, as you might discern if read my regularly updated original article, 8 ÷ 2(2 + 2) = 1: Why PEMDAS Alone Is Not Enough | Sunday Morning Greek Blog or any of the other several articles I’ve written on the subject. But as I feel like I’ve addressed most of the obvious weaknesses of the ignorance of the priority of juxtapositional binding over extant operational signs, those moments of inspiration (dare I say “genius”?) are getting farther apart.
But this new concept I’ve described (I’m not sure how “new” it is, but I know I was never taught anything like this in math) as it turns out is really another way of double checking your division homework. I had gotten close to this when I demonstrated that the PEMDAS crowd’s way of interpreting 8 ÷ 2(2 + 2) was equivalent to the expression 8 ÷ [2/(2 + 2)], the latter of which is subject to “invert and multiply” to solve and therefore more naturally solved without needing to remember or manipulate PEMDAS. [Note: Copilot called this concept “dual symmetry” in division.]
What Stocking’s Reciprocation does is move the emphasis from the divisor to the dividend, thus eliminating the argument about juxtaposition all together, because the answer you get when you double-check with Stocking’s Reciprocation agrees with the answer you get when you consider the juxtaposed divisor to be a single, unbreakable unit or monomial. You can’t get to the PEMDAS crowd’s answer of 16 using Stocking’s Reciprocation, so that proves the answer 16 is not correct and that the PEMDAS crowd’s method of working that expression is greatly flawed.
Here’s how Stocking’s Reciprocation works. Instead of inverting and multiplying the divisor (especially in this case where the divisor is disputed), we invert the dividend and multiply. This will yield the reciprocal of what the correct answer should be.
8 ÷ 2(2 + 2) becomes (1/8) x 2(2 +2).
You can already see how this is going to go then. Now, you have eliminated the obelus and are left with a fraction (which is NOT an explicit division problem) multiplied by a number multiplied by a value in parentheses. With all three factors of the expression multiplied now, it doesn’t matter what order you multiply them (Commutative Property)!
At this point, we can still apply undisputed PEMDAS principles by calculating the value in parentheses first and then multiplying straight across.
(1/8) x 2(4) becomes (1/8) x 8 (remember, order doesn’t matter at this point because it’s all multiplication), so the product is 1. Then you take the reciprocal of the product to compensate for the initial reciprocal conversion, and of course you get 1.
Just to demonstrate this isn’t an anomaly because the answer was 1, let’s do a couple more complicated expressions, as I did on my Facebook response today with one of these expressions:
8 ÷ 4(6 + 8) would be incorrectly calculated by the PEMDAS crowd as 28. But using Stocking’s Reciprocation, you get:
(1/8) x 4(6 + 8) = (1/2)(14) = 7. The reciprocal of that (and the answer to the original expression using Stocking’s Order, i.e., juxtaposed multiplication takes priority) is 1/7. If the 4(6 + 8) is inseparable in that expression, as I have claimed all along, you solve the 4(6 + 8) first, yielding 56, so in the original expression, you would get 8/56 = 1/7. Voila!
So let’s apply it to this expression.
48 ÷ 2(9 + 3) = (1/48) x 2(12) = 24/48 = 1/2, and the reciprocal of that is 2.
Working the problems backwards like this (or is it sideways?) and demonstrating there is no confusion about what the answer is demonstrates that the more sophisticated take on Order of Operations that engineers, physicists, and other math-heavy professionals use daily is 100% correct. They understand instinctively that the juxtaposition means you don’t separate the terms by an extant operational sign. After all, they passed arithmetic and moved on to algebra where that principle is prominent (i.e., 8 ÷ 2a = 4/a, NOT 4a).
I’m going to brag on myself a bit too here. When I posed this concept to Copilot (Microsoft’s AI), I got the following response: “It’s not commonly taught, but it’s mathematically sound and elegant.” Copilot even used my own Facebook post that I’d just put up less than a half-hour earlier, as a source for its analysis.
I preached this message on July 27, 2025 (Year C, Proper 12), at Mount View Presbyterian Church, Omaha, NE.
The last line of our Declaration of Independence says this:
“And for the support of this Declaration, with a firm reliance on the protection of divine Providence, we mutually pledge to each other our Lives, our Fortunes and our sacred Honor.”
The Declaration of Independence uses several different terms for our heavenly Father: “Nature’s God,” “Creator,” “Supreme Judge of the world,” and in the last line of the Declaration, “Providence.” Even “Prudence,”[1] might be considered a reference to God because it derives from the same root for “Providence.”
“Providence,” when capitalized, means “God conceived as the power sustaining and guiding human destiny.” The adjective “provident” is relevant to our passage today: “making provision for the future.”
As you listened to the New Testament reading this morning, you may have thought the switch from an abridged version of the Lord’s prayer to teaching about God’s providence seemed a bit “abrupt.” But if you look closely at how the portion of the Lord’s prayer is formatted in most English Bible, you’ll see that the central line is “Give us this day our daily bread.” He then goes on to relate Jesus’s teaching about what a good neighbor might do even though they might feel “put-out” by the expectation in the middle of the night.
To understand the story of the neighbor’s request, I want to remind you about what I talked about a couple weeks ago about what it meant to be a neighbor and part of an in-group. First of all, notice that Jesus puts his listeners in the position of the one needing the bread. To make this easier to follow, I’m going to give the three men in this story names. Since the man in story is played by you in Jesus’s parable, you can insert your own name in there, but I’m going to use Joel for our example, because he’s always willing to play along with me when I ask for audience participation, and Joel is a good biblical name. We’ll call Joel’s neighbor Hosea since that’s the Bible book that comes before Joel, and we’ll call Joel’s visitor Malachi, since that’s the last book of the minor prophets.
Even though Hosea probably doesn’t know Malachi, because both men are friends with Joel, Joel would have a cultural, in-group expectation that Hosea would show the same hospitality and respect to Malachi as he would to him and would not make Joel look bad to Malachi. The other thing to consider here is that Joel seemingly has enough trust in his relationship with Hosea that waking him up at midnight for bread doesn’t seem out of bounds with the cultural norms of the day. Hosea is certainly not happy about being awakened in this manner, but he realizes that his neighbor Joel must really be in a bind if he has the “shameless audacity” to wake him up at midnight. It is that trust in the relationship between Joel and Hosea that compels Hosea to get up and give Joel the bread he needs.[2]
One final note here: Joel’s audacious act would have signaled to Hosea that if Hosea was ever in a similar spot, Joel would be willing to help him as well. This was not so much a matter of tit-for-tat; Hosea would not have had an expectation of any compensation or immediate reciprocation. It was more of a “pay-it-forward” act. Nor was it “charity” or even “welfare” in the way we might typically think of it. It’s a two-way street. It is simply the mindset of those first-century listeners who would have understood the cultural dynamic at work here and who take their cultural obligations seriously.
The important consideration here is that this cultural dynamic was intended for the success and longevity of the community. When those in the in-group and allied with the in-group are supported by the in-group, the in-group becomes stronger and more closely united. This provides a sense of internal security for the in-group. They don’t have to rely on outside sources to support their cultural priorities.
This is the kind of thing we see going on in the early chapters of the book of Acts. We see the believers willingly selling property to help meet the needs of less fortunate believers. We see them sharing meals together. We see them meeting together for prayer, worship, and instruction. We see them resolving conflicts so that all the widows can be cared for. And we even see Paul condemn Ananias and Sapphira to death for lying about what they gave for those needs, even though they had absolutely no obligation to give every last drachma of their proceeds on the sale.
As I said a couple weeks ago, this kind of internal support system is how the community maintained their honor and integrity. This also represents how God works with us, especially when it comes to prayer. Verses 9–10 give us the moral of the story: don’t be afraid to ask—and be persistent when you do!
I wish I understood why our current culture has gotten to the place where we are afraid or embarrassed to ask for help from those we know, especially within the church. Part of it may have to do with the government thinking they need to take over the responsibility of “charity,” and then regulating it to the point where it becomes burdensome for private nonprofit entities to respond to needs. I fear it’s created a mindset that government should be the first place you look for help, not the church.
But I digress. The last part of our gospel passage this morning reveals the goodness of God. Some may have the opinion that God only wants to punish us, or that the bad things that happen to us are the result of a cruel God. But God is not cruel in the way he responds to our prayers. James confirms the goodness of God as revealed in the last part of our gospel passage:
James 1:5–8 says:
5 If any of you lacks wisdom, you should ask God, who gives generously to all without finding fault, and it will be given to you. 6 But when you ask, you must believe and not doubt, because the one who doubts is like a wave of the sea, blown and tossed by the wind. 7 That person should not expect to receive anything from the Lord. 8 Such a person is double-minded and unstable in all they do.[3]
And later in 4:1–3, James says almost the same thing, with a different twist:
What causes fights and quarrels among you? Don’t they come from your desires that battle within you? 2 You desire but do not have, so you kill. You covet but you cannot get what you want, so you quarrel and fight. You do not have because you do not ask God.3 When you ask, you do not receive, because you ask with wrong motives, that you may spend what you get on your pleasures.[4]
Our reading form Psalm 138 this morning also affirms the goodness of God when we ask him for help. Verse 3 says, “When I called, you answered me;
you greatly emboldened me.[5]” There’s that boldness we saw in the man who asked for bread from his neighbor. Verse 6 says, “Though the Lord is exalted, he looks kindly on the lowly; though lofty, he sees them from afar.”[6]
Your presumed standing with God matters not. In fact, he has a special place in his heart for those of us who think we’re “lowly.”
David is willing to bow down to God and give him unfettered praise for the way he has provided for him. He recognizes God’s protection, God’s preservation, and God’s power. God is there for us when we need him. Even if we don’t get the answer we want, we, like the three men in the fiery furnace, know that God will save us one way or another when we put our trust in him.
Paul affirms this in Colossians 2:6–7 as well: “So then, just as you received Christ Jesus as Lord, continue to live your lives in him, 7 rooted and built up in him, strengthened in the faith as you were taught, and overflowing with thankfulness.”[7] We can show the same thankfulness that David did in Psalm 138.
At the beginning of my message this morning, I mentioned how the Declaration of Independence refers to God as “Providence.” He is the provider and we can trust in him. But this also implies a mutual trust as well, just as we have as the body of Christ united. Jefferson closes the Declaration of Independence with a line that sums up nicely the commitment we should have to the body of Christ as we walk faithfully with Jesus: “We mutually pledge to each other our Lives, our Fortunes and our sacred Honor.”[8] We are God’s witnesses on earth to his great and marvelous deeds. How can we do anything less than give him all our praise for the ways he provides for us. Don’t be afraid to ask God to do some great things for you as you walk with him. Amen.
[1] According to Merriam-Webster, “prudence” is derived from the same root as “providence.” “Prudence” means “the ability to govern and discipline oneself by the use of reason” or “skill and good judgment in the use of resources,” which is perhaps a meaning relevant to this message.
[2] This is known as a “dyadic alliance” or “dyadic contract.” See Malina, Bruce J. The New Testament World: Insights from Cultural Anthropology, revised edition, pp. 99–103, 115. Wesley/John Knox Press, 1993. See also Malina’s Windows on the World of Jesus, pp. 48–49.
I preached this message July 13, 2025, at Mount View Presbyterian Church in Omaha, NE.
Good morning. The Lord be with you.
I imagine that most of us either grew up watching Mr. Rogers’ Neighborhood or had young children who watched that show. (It’s okay to admit it if you watched it with your kids.) As a kid, one of my favorite parts of the show was when he sang “Who Are the People in Your Neighborhood?” He would usually find someone who could demonstrate their talents or skills. When I watched it with my kids later on, that had become a visit to a factory where something was made. I was fascinated by the episode on how aluminum foil was made.
Of course, being a train lover, my other favorite part of the show was when Trolley went to the Magic Kingdom. I never thought much about it as a kid, but as a parent it dawned on me, spoiler alert here, that Mr. Rogers was doing most if not all of the puppet voices.
As a kid, it was just a fun show to watch. But as I watched it as an adult with my kids, especially in the small rural town we lived in in Illinois and where I served as a pastor, I began to broaden my concept of who my “neighbors” were. I used to think “neighbors” were just the people who lived in the houses around us. But as I would go for walks with the kids or go to the park with them, we would meet people we didn’t know or I would meet the parents of my kids’ classmates.
If I said hi to someone in passing, my kids sometimes asked, “Who was that?” If I didn’t know their name, I would usually say, “That’s one of our Paxton neighbors.” After all, they may have been neighbors to people in our congregation, so I wanted to make sure they knew that the newcomer to the town wanted to fit in.
In Mediterranean culture, the concept of neighbor had a slightly different nuance. Their culture was steeped in the concept of “in-group” vs. “out-group.” The concept of neighbor went much further than just people who live near you. Bruce Malina, a Theology professor who taught for 48 years at Creighton before his death in 2017, defined “neighbor” in the Bible this way:
“The term refers to a social role with rights and obligations that derive simply from living close with others—the same village or neighborhood. Neighbors of this sort are an extension of one’s kin group.”[1]
In other words, neighbors were family and were to be treated like family. Along with that, being considered family meant you had a certain degree of honor in the community as well. But that wasn’t a hard-and-fast rule for Jews. If you did something to violate the honor or trust you had in the community, you would most likely be shamed or even ostracized. If you got drunk and embarrassed yourself or got in a fight or damaged someone else’s property, the community would not excuse or overlook such behavior. If you stole from your neighbor or committed adultery, you got more than a slap on the wrist. You would find yourself in the out-group.
These concepts of honor and shame, in-group and out-group, have lost their force in modern culture. Malina documents that in one of his other books.[2] In fact, in some respects, this has been flipped on its head. The in-groups value law and order in their communities and have certain expectations about what good behavior looks like, while the out-groups, something that used to carry shame and guilt, are now intentionally and sometimes violently trying to disrupt the law and order and care little about good behavior or even trying to be restored to an in-group.
It’s not that there’s only one in-group for everybody either. Several in-groups peacefully coexisted in neighboring communities and were distinguished by any number or combination of cultural and ethnic characteristics. But they all had the overarching concept of honor and shame and could peacefully interact with other in-groups when they acted honorably and without malice. If anyone acted with malice or violence toward another in-group, the gloves were off at that point.
This is some of the cultural background that was assumed by the authors of the Bible when they wrote. We in America tend to read the Bible through our 21st-century cultural lens and will sometimes get a little uncomfortable with the way the Bible describes a certain scenario. That’s because the scenario is set in a time and place with a completely different worldview. This Mediterranean worldview is the lens through which we should view the Parable of the Good Samaritan.
One caveat, I’m not excusing anyone’s behavior in the story. Jesus clearly expects the expert in the law to come up with the only right answer to the question he asked after Jesus finishes the story. Yet we’re somewhat uncomfortable with the fact that servants of God bypassed the wounded man in the parable.
First off in the story, we notice that the man is not identified by any ethnic or cultural features. In the story, he’s just a man. He also appears to be traveling alone. The path from Jerusalem to Jericho was a bit of a challenge, a descent of nearly 3,400 feet over 17 miles, so that’s a 200-foot change in elevation every mile through rocky terrain. Plenty of places for bandits to hide.
Priests and Levites were generally respected in that day, and they were easily recognizable as well by the garments they wore. As such, many people would have looked up to them, even those people who were not Jewish, so they would not have typically been targeted by bandits. It’s not that the bandits cared about their reputation with any one group; they just knew that the civil and legal penalties for them would have probably been a lot more severe.
For the priest and Levite, those jobs were their livelihood and gave them a certain social standing within their “in-groups.” When each of them walked by the bloodied-up man on the side of the road, their first thought wasn’t to help. There’s no indication in the story that either of them even bothered to check if the man was Jewish himself. No, their first thought was, “If this man is dead, I’m going to be unclean and not able to do my job.” They had a legalistic view of the law that they thought they could use to protect their “status” in their in-groups, but we all know by now that Jesus wasn’t interested in the legalistic interpretation.
The people hearing the story, including the expert in the law who asked the question, probably expected that would be the behavior of the priest and the Levite. But then Jesus throws a twist into the story that the expert may not have been expecting. He says a Samaritan came along and helped the man thoroughly, even using his own money to pay for his care until he could return. Now Samaritans were definitely not part of the Jews’ in-group. So when Jesus asks the expert to identify which one was the neighbor, you can imagine that the expert must have gulped a bit and tugged at his collar uncomfortably.
Since the Jews were known for walking around Samaria rather than going through it, which was the more direct route, one could say here that the priest and the Levite treated the wounded man like a Samaritan, even though they probably didn’t know if the man was a Samaritan. If the man was a Jew, then they actually violated a cultural norm that was probably considered to be on the same level with the Law.
Meanwhile, the Samaritan, who doesn’t hate the Jews as much as the Jews hate them, chose to treat the man like he would treat his own family, without concern for whether the man was Samaritan, Jew, or some other ethnic background. In other words, as the law expert correctly discerned, the Samaritan treated the man like a neighbor.
What Jesus is obviously getting at here then is that being a neighbor or showing a stranger the same level of respect as a neighbor isn’t a noun or an adjective, it’s a verb. It doesn’t matter what your background is; you’re considered a neighbor by the way you treat those you come in contact with. That doesn’t mean you’ve abandoned your primary in-group and joined the stranger’s in-group, nor does it mean you’ve adopted or supported any beliefs or customs of the stranger’s in-group contrary to your own. When you act like a neighbor, you’re not “guilty by association,” at least not from Jesus’s perspective.
In verse 29, Luke tells us the legal expert wanted to “justify himself” by asking the question. Do you suppose the legal expert accomplished that? I’m guessing not, but we’ll never know. Just because Jesus tweaked the definition of neighbor here doesn’t mean he overturned the idea of one’s own “kinship” in-group, though. The question each of us must ask, then, is the same question Mr. Rogers sang in his show: “Who are the people in YOUR neighborhood?” Not only that, but “How are you loving the people in your neighborhood?” You may have guessed by now that when I say “neighborhood,” I don’t just mean inside these four walls.
In an age where we’re personally and increasingly isolated by either technology or mobility, it’s important that we have a neighborhood to connect with so we can feel the support, love, and encouragement that comes from those associations. I pray that you might get to know a new neighbor this week and share the love of the savior with them. Peace to you. Amen.
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[1] Malina, Bruce J. Windows on the World of Jesus: Time Travel to Ancient Judea. Westminster/John Knox Press, Louisville, KY, 1993, p. 52.
[2] Malina, Bruce J. The New Testament World: Insights from Cultural Anthropology. Revised Edition. Westminster/John Knox Press, Louisville, KY, 1993, pp. 82–86.
My opinions are my own, except where otherwise cited.
This article was inspired by a response to a reader’s comment on one of my other PEMDAS/OOO articles. I thought the point I made was worthy enough to be a separate post.
The idea that a fraction is identical to a division problem with an obelus just doesn’t hold water if you analyze the uses of fractions. For example, if I say I ate two-thirds of an apple pie, I’m not intending that to be a division problem. It’s a ratio indicating how much of the pie I ate, i.e., it’s a stand-alone value, or a monomial. We must consider as well that a fraction is more accurate than a decimal if the decimal value has a repeating pattern. Perhaps you’re familiar with the proof that 0.999999…. (repeating to infinity) equals 1.0. If 1/3 (0.333…..) + 2/3 (0.666…..) = 1.0, then the sum of the decimal equivalents must equal 1. Via mathematical proof, this is proven true. However, you never actually get to the point of being able to “carry the 1,” so the decimal equivalent of the sum of the fractions is 10^(-∞) off.
Granted, such a difference in real measurement is not humanly perceptible, but it does reveal a potential issue when talking about astronomical or microscopical measurements, for example. Even if the fraction is part of a larger expression or formula, it’s always possible that the denominator of the fraction will “cancel out” exactly as a common factor with another element in the expression, so the simplification process is the “division,” but it may not be with the numerator of the fraction. For example, if I have the expression 6 x (2/3), I “cancel” the common factor of three from the 6 (leaving 2) and the denominator 3 (leaving 1, thus a whole number), so I’m left with 2 x 2 = 4. Division is happening in the cancellation of the common factor, but it happens with an element outside of the fraction itself. Therefore, the fraction itself is NOT a division problem. There are other elements acting on the fraction.
What is key to the monomial discussion, then, is the juxtapositional binding. In the expression 8 ÷ 2(2 + 2), for the sixteeners to claim that somehow the 8 ÷ 2 becomes a fractional coefficient is contrary to the concept of the fraction. The obelus doesn’t have the binding property inherent with the fraction construction, so to replace the obelus with a fraction bar and say “they’re grouped” while unbinding the 2 from the (2 + 2) defies the clearly indicated juxtapositional binding. They try to have it both ways, but for whatever reason, they’re not seeing that. (Personally, I think they got burned at some point thinking they were justified in adding the multiplication sign and they got corrected by someone who believes as we do, and instead of thinking it through, they hunkered down in their rebellion, but that’s another story.)
This is where the linguistic aspect comes into play. We’ve seen how Wolfram interprets “Eight divided by twice the sum of two plus two” as 1, but “Eight divided by two times the sum of two plus two” as 16. “Twice” and “two times” mean the same thing at face value, but “twice” binds to the “two plus two” as an adverbial modifier, while “two times” is treated as a subject-verb combination. The latter treats “two” as the subject and “times” as the verb substituting for the multiplication sign, thus 2 x (2 + 2). “Twice” has the multiplication implied in the adverbial phrase “twice the sum of two plus two,” so this more accurately reflects the implied multiplication of 2(2 + 2). As such, 2(2 + 2) is the mathematical equivalent of an adverbial phrase intended to be taken as a semantic unit, with or without the external parentheses, just like a fraction by itself is not a division problem, but a singular or monomial value. Therefore, 8 ÷ 2(2 + 2) = 1. Period. End of debate.
In today’s educational environment, there is a lack of critical thinking development. Young people are trained in mathematics simply to get to the point of an answer, not to learn the theory behind mathematics. In other words, they’re trained like calculators, and in this instance, they’re trained to process a problem like a cheap calculator would solve it instead of taking a broader view of the linguistic aspects of any mathematical expression. It’s really no different than giving a non-English speaker an English text and English dictionary and asking them to translate without any concept of grammatical rules, syntax, idioms, etc. PEMDAS/OOO is a formula designed to follow how a calculator solves an expression, not how students of math have solved the expressions in the past and certainly not to make the paradigm consistent with algebra, which treats an expression like 8 ÷ 2a as 8 ÷ (2a) and NOT 8 ÷ 2 x a. The linguistic argument supports the algebraic view of such expressions and therefore supports the same view for arithmetic.
I preached this sermon on June 29, 2025, at Mount View Presbyterian Church. This is “Proper 8” (third Sunday after Pentecost) for Year C of the Lectionary. For future reference, the next time these passages will appear in the Lectionary together is July 2, 2028.
Good morning! The Lord be with you!
How do you follow a man who says he’s going to die? Some might ask, “What kind of man would ask someone to follow him knowing that he’s going to die?” But is that the right question? I mean, we’re all going to die at some point, and we don’t know when. Maybe the question should be, “What does the man who’s going to die think about the person he’s asking to follow him?”
I’m sure the would-be follower would have questions for Jesus as well. “Why would you choose me? I’m just a fisherman.” “What can I expect from following you? You don’t look like you have much.” Or maybe the question is a little more self-reflective: “What does he see in me that I don’t see in myself?”
These types of questions are, to a certain extent, somewhat academic or rhetorical. I think most of us realize, and can see plainly in the gospels, that Jesus taught like no other and that he worked miracles like no other. These two features of his life on earth were undoubtedly the most attractive features of his ministry. They were also the focus of Luke’s gospel from the account of his miraculous birth and the miracles that surrounded that up to the point of our gospel passage this morning.
Note what Luke says about Jesus’s mind set in the first part of vs. 51: “When the days drew near for him to be taken up.” Jesus knew he was going to die in Jerusalem at the hands of the chief priests and religious leadership. But Luke suggests Jesus also knew here that whatever that death entailed, he would defeat it and win victory over the grave, because his resolve, apparently, comes from his divine knowledge that he would defeat death and be resurrected to return to God.
Our reading in the bulletin this morning from the New Revised Standard Version gives the literal translation of the Greek in the last part of verse 51: “Jesus set his face to go to Jerusalem.” Other translations are more descriptive with that idiom. The New International Version says, “Jesus resolutely set out for Jerusalem,” while the New American Standard Version says, “He was determined to go to Jerusalem.” This is remarkable since Jesus twice predicts his death in the earlier verses of Luke 9—once before his transfiguration and once after it. Jesus knows the end result and is still steadfast to his ultimate mission.
It is interesting, then, that in the context of Jesus showing such resolve that Luke tells us about several others who want to follow Jesus as well. His disciples were already on board with all this, except perhaps for Judas. But as much as Jesus wants people to follow him, now is not the time for the feeble of heart or the weak-willed to be following a man who is resolutely going toward his death. I would imagine Luke picks a few representative samples out of the dozens, if not hundreds, who want to follow him at this point. His answers may seem a bit terse or harsh to us, but he wants any potential followers to understand just what they’re getting themselves into. There is no turning back once you’ve decided to put the hand to the plow.
Our reading from Psalm 16 this morning seems to fit the bill as encouragement for someone who is determined to follow Jesus. David calls this psalm a miktam. Only five other psalms have that designation—Psalms 56–60—and most of them have to do with the author appealing for refuge or deliverance from their enemies. David expresses a great deal of confidence in his relationship with God in this psalm. From God’s guidance and instruction to his provision of joy and even, it would seem, to the promise of resurrection from the dead: “You will not abandon me to the realm of the dead.”
We can break this Psalm down into three parts, at least that’s how my NIV interprets it. After an opening plea for refuge, the next three verses reveal that David has confidence in those who are holy, that is, in his own community. Because of his confidence in them, not only can he praise God, but he can also confidently assert he has no intention of following other gods. He knows that following other gods would only bring pain and suffering to him and leave him without hope.
In the next four verses, David praises the Lord for being his provider and defender. David feels secures because he has three things from God in this regard: boundaries, instruction, and a focal point. Boundaries are good for us, because they tell us what the limits are and where we need to be to stay safe. A study many years ago showed that children felt more freedom and security to explore and move around a playground when it was fenced in. If a ball got away from them, they were less afraid to chase it to a hard boundary, especially if such a boundary was next to a busy street or other potential hazard. A boundary can work both ways as well. A playground fence also helps to keep stray animals and people with malicious intent from easy access to the area.
Jill and I were in the mountains last week in Colorado. Of course there are boundaries everywhere there. And not just physical boundaries. The twisty road through the canyon had guard rails at places to prevent vehicles from going into the river. We had to observe the speed limit signs for tight and blind curves so we wouldn’t run into the big horn sheep or the bicyclists we encountered on the road. A double yellow stripe down the middle of the road with a rumble strip let us know if we were straying into the other lane.
God’s boundaries are found in the instructions we have in Scripture, namely the Ten Commandments, and the two greatest commandments Jesus reminded us of: love God with all your heart, soul, strength, and mind; love your neighbor as yourself. Following those boundaries can keep us from all kinds of evil consequences. More on this in a moment.
Verse 8 reminds us of Hebrews 12:1–3:
Therefore, since we are surrounded by such a great cloud of witnesses, let us throw off everything that hinders and the sin that so easily entangles. And let us run with perseverance the race marked out for us, 2 fixing our eyes on Jesus, the pioneer and perfecter of faith. For the joy set before him he endured the cross, scorning its shame, and sat down at the right hand of the throne of God. 3 Consider him who endured such opposition from sinners, so that you will not grow weary and lose heart.[1]
When we keep our eyes fixed on the risen savior, we can always see the end result: our resurrection and eternal home with him in glory. In spite of the boundaries and instruction God gives us, we also know there will be times when our faith is challenged: the loss of a loved one, a critical medical diagnosis, broken relationships, etc. But as David says, we can stand firm and not be shaken.
In the last three verses, David speaks of his confidence as he considers his own future death. Even before he has the example of our risen savior he seems to understand the concept of resurrection: “You will not abandon me to the realm of the dead, nor will you let your faithful one see decay.” We too can have this confidence when we follow Jesus and honor him as Lord of our lives. Eternal rewards await us, and God has graciously made that available to us through the death and resurrection of his son.
We are not without a more practical example in the New Testament of the principles David laid out in Psalm 16. The other New Testament passage from the lectionary today is from Galatians 5. Paul talks about the difference between living a life without boundaries and a living a life bound to the Holy Spirit that Jesus imparts to all believers.
It is for freedom that Christ has set us free. Stand firm, then, and do not let yourselves be burdened again by a yoke of slavery.[2]
…
13 You, my brothers and sisters, were called to be free. But do not use your freedom to indulge the flesh; rather, serve one another humbly in love. 14 For the entire law is fulfilled in keeping this one command: “Love your neighbor as yourself.” k15 If you bite and devour each other, watch out or you will be destroyed by each other.
16 So I say, walk by the Spirit, and you will not gratify the desires of the flesh. 17 For the flesh desires what is contrary to the Spirit, and the Spirit what is contrary to the flesh. They are in conflict with each other, so that you are not to do whatever you want. 18 But if you are led by the Spirit, you are not under the law.
19 The acts of the flesh are obvious: sexual immorality, impurity and debauchery; 20 idolatry and witchcraft; hatred, discord, jealousy, fits of rage, selfish ambition, dissensions, factions 21 and envy; drunkenness, orgies, and the like. I warn you, as I did before, that those who live like this will not inherit the kingdom of God.
22 But the fruit of the Spirit is love, joy, peace, forbearance, kindness, goodness, faithfulness, 23 gentleness and self-control. Against such things there is no law. 24 Those who belong to Christ Jesus have crucified the flesh with its passions and desires. 25 Since we live by the Spirit, let us keep in step with the Spirit. 26 Let us not become conceited, provoking and envying each other.[3]
Notice the contrast Paul makes here: He speaks of the “acts of the flesh,” which refer to behaviors without boundaries. He warns that theses kinds of behaviors can lead to us “biting, devouring, and destroying” each other. Not only that, Paul also warns that people who live wantonly without boundaries “will not inherit the kingdom of God.” There’s no wishy-washy language here. You’re either in or out.
Contrast the “acts of the flesh,” then, with the “fruit of the Spirit.” Paul doesn’t mention behaviors here but rather a mindset by which to live. They are boundaries that come from being filled with the Holy Spirit. They put a check on our behavior and keep us from flying off the handle when we get angry or unduly criticizing someone without understanding the background of a situation. They help us to remember that God loves people first and foremost, regardless of where they’re at in their faith journey. But they do not absolve us of the responsibility to share the good news either.
As a preacher, I could craft a whole nine-week sermon series on the fruit of the Spirit, so I don’t even want to try to expound on that here. I’m sure most of you have been around long enough that you understand those character qualities quite well. Titus 2:11–14 gives a nice summary, however, and I’ll wrap things up with this passage this morning:
11 For the grace of God has appeared that offers salvation to all people. 12 It teaches us to say “No” to ungodliness and worldly passions, and to live self-controlled, upright and godly lives in this present age, 13 while we wait for the blessed hope—the appearing of the glory of our great God and Savior, Jesus Christ, 14 who gave himself for us to redeem us from all wickedness and to purify for himself a people that are his very own, eager to do what is good.[4]
Let me return to my original questions, “How do you follow a man who says he is going to die?” You follow him, Jesus, because you know he won victory over death and the grave and wants to share that victory with you. “What does God see in me that I don’t see in myself that he wants me to follow him?” He knows and sees the power of transformation the Holy Spirit can work in those who choose to follow Jesus. If you need a reminder of that, just pray and ask God to refresh and renew your experience with him. He desires all to be saved; any excuse you may have to not follow is not enough for God to give up on you. May God strengthen your faith and refresh your hope as you go from here today. Amen!
Historical context and notes: I preached this message on Father’s Day, June 15, 2025, at Mount View Presbyterian Church. It was also “Trinity Sunday” on the Lectionary calendar, the Sunday after Pentecost. Culturally, this weekend also saw the parade/celebration for the 250th anniversary of the U.S. Army, “No Kings” protests nationwide (thus a few extra references to God as our King), the onset of a conflict between Israel and Iran with the goal of degrading Iran’s nuclear weapons capabilities, and the politically motivated assassination of a Minnesota State representative.
Happy Fathers’ Day, and a blessed Trinity Sunday to all of you. I’ve had a busy three weeks traveling hither and yon. Two weeks ago, my brother and I took our third annual fishing trip to South Dakota and caught our limit of walleye both days. Last weekend, my wife and I went to Branson with our small group and saw the production of David at the Sight and Sound theater. If you ever go to Branson, the Sight and Sound theater is well worth the price of a ticket. The entire operation is a ministry that focuses on telling biblical and patriotic stories. They tie the biblical story to the message of the cross toward the end, and after the show some of the cast make themselves available to pray with people.
Now even though I had a very relaxing few weeks off and feel somewhat refreshed from a busy schedule, I’m not ashamed to admit that it’s been kind of tough to focus on writing a message this week with all the other chaos going on in the world. Nevertheless, I think perhaps the example of Jehoshaphat in the Old Testament can help us deal with the potential chaos some may be experiencing. When Jehoshaphat was faced with a nearly impossible battle in 2 Chronicles 20 against the Moabites, Ammonites, and Meunites, he had the people pray in the temple courtyard. The next morning, he put the men’s choir out in front of the army as they marched toward the Desert of Tekoa. As they sang, God set ambushes, and the three opposing armies wound up destroying each other. Israel never had to lift a finger to fight. God is the true King, and when we put him first, good things can happen.
In our Old Testament reading this morning, the first couple verses of Psalm 8 say this:
1 Lord, our Lord,
how majestic is your name in all the earth!
You have set your glory
in the heavens.
2 Through the praise of children and infants
you have established a stronghold against your enemies,
If those two examples aren’t enough to show the power of praise, consider the story of the walls of Jericho which, by the Jews marching around the city, blowing their trumpets, and lifting up a mighty shout of praise, crumbled as a result of that sonic boom. God is the true King, and when we put him first, good things can happen. Psalm 22:3 in the English Standard Version says, “Yet you are holy, enthroned on the praises of Israel.”[2] If you’re used to the King James Version, that verse is translated, “Thou that inhabitest the praises of Israel.”[3]
This demonstrates the point that our praise is mighty and effective against evil because the God who dwells in that praise as it goes forth from our lips and our lives is mighty. This is God Almighty, God the Father, God the King, the creator of all that is made, even life itself; the giver of breath; the author of wisdom and truth.
We see the might, power, and even the orderliness of God in the creation narrative, for example. On Day One, he begins with the “formless and void” rock we call Earth and creates “light.” We don’t know what that light is, because the things that make or reflect light aren’t created until Day 4. What is this Day One light then? Is it the light that emanates from God because of his spiritual nature? Is it the afterglow of a “big bang” that produced the formless and void Earth and everything else in the universe? Is it meant to have a more metaphorical meaning like moral clarity? Or is it a reference to someone who would later claim that he is the light of the world, and of whom John would make the claim that nothing in this world was made without him? Hmmm. More on that later.
On Day Two, God separated the waters below from the waters above, most likely a vapor canopy that created a greenhouse effect for the new life that was coming. The separation between the waters was called “sky.” On Day Five, he created the creatures that dwell in the air and the creatures that dwell in the sea.
On Day Three, he brought forth dry land and created the diversity of flora that grows on the earth today. I find it interesting that the plants that need the sun to photosynthesize and grow are created the day before the sun is created. Anyone ever notice that? That’s one reason why I think each of the days of creation represent a 24-hour time period. If the vegetation had been created thousands of years before the sun came to exist, it would not have survived. On Day Six, he creates all the creatures that would dwell on land, including his ultimate creation, Mankind.
With each day of creation, not only is God creating people and animals and plants and objects in the solar system, he’s also creating all of the physical, biological, geological, and psychological (and all the other “-logicals”) rules and principles by which all the natural, or created, world operates under. On Day Seven, God rested. He had taken the “formless and void” third rock from the sun and transformed it into a well-formed, orderly creation.
Not only was God mighty and powerful as our king, though. He was also the epitome of righteousness. In him there is no fault, no stain, no sin. Perhaps that is why his “light” is the brightest of all, so bright that no mortal, sinful man can stand in the presence of it. From the Fall to the Flood, God demonstrated great patience with the wickedness of man, but God had a built-in judgment plan. The vapor canopy had worked quite well to ensure the young earth would flourish and grow, but man’s wickedness had become too much for God to bear. He told righteous Noah to build an ark and brought Noah a pair of every kind of animal to rescue them through the Flood.
The Bible says the rain came down and the flood gates of the earth were opened. Sounds to me like a giant meteor pierced the vapor canopy and all that water condensed and fell to the earth. It also may have broken up Pangea, the not-so-hypothetical single continent that once existed on Earth and started what we know today as plate tectonics, the movement of the continents, and all the fun stuff that comes with that, like earthquakes and volcanoes. This shows the enduring power of God’s creation, but it also shows that he is a God who expects the praise we give him for his righteous judgments.
The signature expression of God’s righteousness is the Ten Commandments. The first three commandments are specific to our direct relationship with God: Don’t put anything above God. Don’t make an image of God to worship. Don’t misuse the name of God. I think we all get that. The next two are positive commands that have to do with what God expects from us: Keep the Sabbath day holy. Honor your parents, who represent God’s authority over you on earth.
The final five commands have to do with our relationships with one another. They are prohibitions against committing violent acts. According to Merriam-Webster, “violence” not only means committing a physical act of aggression like murder, assault, or rape, but it can also mean “injury by or as if by distortion, infringement, or profanation” and gives the synonym “outrage.” Even coveting is not just a thought crime about desiring someone else’s property. When Jesus summarizes the Ten Commandments in Mark 10:19, he uses the Greek word translated “defraud,” “cheat,” or “rob” in the New Testament (ἀποστερέω apostereō). Coveting is violence, because its goal is to obtain something by illicit means.
Because the Ten Commandments are God’s foundational laws, and because they addressed fundamental issues of our relationships with God and with others, a violation of any of them could have resulted in the death penalty, were it not for the provisions in the law for blood sacrifice and the forgiveness of sin. But God knew from the time of the Fall he would need another way to address mankind’s sin. That’s where the second person of the trinity is introduced to the world.
Of course, this is Jesus, the son of God, conceived by the Holy Spirit, born of the virgin Mary. He would come to walk among us as a human being and learn, through his fully divine nature and insight”, what it was like to live as a mortal among mortals. Hebrews 4:15 says: “For we do not have a high priest who is unable to empathize with our weaknesses, but we have one who has been tempted in every way, just as we are—yet he did not sin.”[4] This Law is good because it shows us what sin is, but the Law itself is not able to provide forgiveness, righteousness, or salvation. Only perfect obedience can do that, but no one is perfect, at least, no one who is fully mortal.
Romans 10:4 says: “Christ is the culmination of the law so that there may be righteousness for everyone who believes.”[5] That word “culmination” (τέλος telos) doesn’t mean the Law is no longer relevant. It means that what the Law was intended to accomplish, imperfect as we are, is now accomplished through the faithfulness of Christ in his death on the cross and our faith in acknowledging Jesus Christ as our risen Savior. This is what John means when he calls Jesus “the word.” What we call “the Ten Commandments” in Hebrew is just simply “Ten Words” (עֲשֶׂ֖רֶת הַדְּבָרִֽים ʿǎśě·rěṯ de·ḇār îm). Jesus’s death on the cross paid the penalty for all time for violating God’s Law. All we need to do is trust in his grace and mercy and live faithfully for him. He is, after all, declared to be Lord of Lords and King of kings in Revelation 19:16.[6]
This is where we meet the third person in the Trinity. I’m not sure what passage your speaker addressed on Pentecost last Sunday, but if it was John 14, you would know that Jesus promised the Holy Spirit would come and teach us what we would need to know to live faithfully for Christ. We do have the Bible, but without the Holy Spirit to help us spiritually understand, discern, and apply the words of the Bible, they are ultimately just words on a page. The Holy Spirit is the divine presence in our lives. The Spirit is the fulfillment of the promise Jesus made at his ascension that he would be with us always even to the end of the age.
The words of our gospel reading this morning bear repeating here:
12 “I have much more to say to you, more than you can now bear. 13 But when he, the Spirit of truth, comes, he will guide you into all the truth. He will not speak on his own; he will speak only what he hears, and he will tell you what is yet to come. 14 He will glorify me because it is from me that he will receive what he will make known to you. 15 All that belongs to the Father is mine. That is why I said the Spirit will receive from me what he will make known to you.”[7]
In a world hounded by chaos and strife in these days, I find it comforting that we have a God who loves us and has provided the way of salvation for us through Jesus’s death and resurrection and the infilling of the Holy Spirit. It is through the Spirit that we can also lift up songs of praise and worship, which brings us full circle this morning. God inhabits the praise of his people by virtue of the Holy Spirit dwelling in our lives. This is our powerful weapon to confront the evil around us. The Spirit also brings comfort, healing, and restoration to our lives.
The words from last week’s gospel passage are appropriate to repeat here, and I’ll close with this.
27 “Peace I leave with you; my peace I give you. I do not give to you as the world gives. Do not let your hearts be troubled and do not be afraid.”[8]
My brother and I went to Lake Francis Case (Chamberlain, SD) for the third time in as many years for what has become our annual walleye fishing trip. I want to give a shout-out[1] to Jason Sorensen, operator of South Dakota Walleye Charters, and Jordan Miles of Hooked Outdoors SD, who piloted the boat and guided us to a great fishing spot near the mouth of the White River. We both got our limit of walleye each day (4/day; one was 20¾”), and my brother hauled in a nice white bass as well. Here are the pictures of our spoils from two days on the boat.
I don’t fish often enough to know where the good spots are, and I wouldn’t necessarily trust Google to provide me that information. In addition, since the walleye like to hang out in about 8–12 feet of water, it’s hard to fish for them from the shore, and neither my brother nor I own a boat. The guide is an economic and practical option for us, then, to get to where we need to go.
The guide also has the necessary tools to find the fish as well. The Garmin technology he had on his boat not only guided us down river in a heavy early morning fog, but it also revealed much of what was hidden underneath us in this mighty muddy Missouri River reservoir. It can map the riverbed and show us where the fish are swimming. Walleye are typically bottom dwellers, so we use “bottom bouncer” weights that keep the bait toward the bottom of the river.
It should go without saying that we all need guides in our journey with Jesus. If you’re a seeker, you have a couple sources of guidance. The fact that you’re seeking some life answers in a relationship with Jesus most likely indicates the Holy Spirit has been prompting you and preparing you for a decision to become a Christ-follower. You also may have a Christ-following friend or acquaintance who has had some influence on you as well. While your friends may understand what is going on in your life and can provide much needed emotional and even physical support, the Holy Spirit knows best what is going on inside your heart and soul, and he knows what is best to provide whatever comfort, assurance, or healing you need on the inside. If you’ve been reading the Bible, both the Holy Spirit and your Christ-following friends can provide help with understanding it if you just ask.
If you are a Christ-follower, then you already know that Scripture, the Bible (aka God’s Word), is our ultimate source of guidance. You already know that you have received the gift of the Holy Spirit upon repenting and being baptized (Acts 2:38). The Holy Spirit will guide you into all truth, but he will never contradict what the Bible says. Reading and studying God’s Word helps to engrain the truths of God’s word into your heart, soul, and mind. Other Christian writers can provide more specific or detailed guidance as well. The stated goals of my blog are to help you “dig deeper, read smarter, and draw closer.” I’m always happy to answer any questions readers may have. If I don’t know the answer, I can usually point you in the right direction.
Experienced biblical scholars usually have a wealth of knowledge about background material relevant to the biblical accounts. They’ve studied the histories and writings of the cultures the main characters of the Bible interact with. They can also help explain some of the background customs and worldviews that are assumed and often unspoken by the biblical authors. Christ-followers who’ve studied in the hard sciences can add insight as well to things like the geography of the day, the geologic history of an area, or other culturally influenced features like architecture, art, and iconography. People trained in medical or mental health practice can also add insight to the wonderful creations we are, physically, spiritually, and socially.
In John 14:15ff., Jesus promises the Holy Spirit and instructs his disciples on what to expect from the Holy Spirit’s infilling and guidance. While the Spirit may speak to those who are seeking God but who are not yet Christ-followers, the Spirit does not dwell in those who have not fully accepted Jesus as their savior. If you are a Christ-follower, then you have assurance of the presence of the Holy Spirit in your life (and you do NOT need to manifest any gifts of the Spirit to prove that!). The Spirit is described as our Advocate in the NIV. Other versions use terms like Comforter, Counselor, or Helper. The Greek word (παράκλητος paraklētos) implies one who is called alongside you. Another role of the Holy Spirit is to remind us of the teachings of Jesus and more broadly the Bible. Jesus also uses the image of “peace” to describe the Spirit’s role in our lives, bringing order in the midst of our chaos; assurance in the midst of our turmoil.
The penultimate promise of Jesus in John 14:27c is this: “Do not let your hearts be troubled and do not be afraid.” We can live in that assurance when we have the peace of Christ dwelling in our hearts. Jesus’s ultimate promise, however, comes in the very next verse: “I am going away and I am coming back to you.” Both of these promises are repeated from the beginning of chapter 14 (vv. 1a, 3). The Spirit is meant for our life on earth. When we get to heaven, our joy and our peace will be to dwell forever with the risen and resurrected savior himself.
Peace to all of you, and thank you for reading.
Scott Stocking
My opinions are my own.
[1] Shout-outs from me do not imply the respective proprietors’ endorsement of my blog. These are a simple courtesy to the proprietors.
Basic mathematics should not have any ambiguity. It’s time to declare juxtaposition as a property of mathematics (and our Arabic number system in general) and silence the ill-informed ambiguists!
The PEMDAS Convention:
Please
Excuse
Me
Doubting
A
Silly
Convention.
Harvard University math professor Oliver Knill has a wonderful site (Ambiguous PEMDAS) that documents many of the arguments against the PEMDAS convention (at least by the PEMDAS/Order of Operations legalists) of treating juxtaposed implicit multiplication with the same priority as multiplication with an extant operational sign. Of course, he’s referring to hotly debated, viral math expressions like:
8 ÷ 2(2 + 2) and 6 ÷ 2(1 + 2),
both of which equal 1 in my book, but to the underinformed PEMDAS/Order of Operations crowd, they equal 16 and 9 respectively.
Similarly, the Math Doctors site has an excellent article entitled Order of Operations: Historical Caveats – The Math Doctors in which they say that “nobody” made the rules for the order of operations and that the so-called “rules” are only descriptive. Doctor Peterson of the Math Doctors has been kind and gracious in answering many of my pointed questions and objections to the normalization of a PEMDAS convention that does not uniformly acknowledge the power of juxtaposition of numerals to imply certain mathematical operations, and for that I am grateful.
But all this begs the question: Why should there be ANY ambiguity in basic mathematics? Why do we have rules that are “descriptive” and not theoretically supported and sound? Isn’t mathematics a hard science (not Barbie’s “Math is hard!” but lacking or minimizing subjectivity), or at least the foundational backbone of the hard sciences that supports the objectivity needed for the hard sciences? Maybe most of the world doesn’t care about this argument about PEMDAS, but I see the admission or acknowledgment or tolerance (or whatever synonym) of ambiguity as a weakness of the discipline. The refusal to acknowledge the inconsistency of rejecting the binding power of juxtaposition in the 2(2 + 2) or 2(1 + 2) part of the expression flies in the face of what is otherwise inherent, intuitive, and obvious about the way the Arabic number system in its current form is constructed.
I agree with the Math Doctors that “nobody” made the rules of Order of Operations and that the “rules” of Order of Operations are descriptive (as opposed to theoretical, which seems to be the implication), but should we really allow such a laissez-faire, passive convention to be immune to such criticisms as I have raised in my writings and videos?
If a mathematical convention has no clear origins or has no theoretical background, then either someone should propose a theoretical background to support it or promulgate a rule based on demonstrated, objective properties and principles (as I have done above) that rightly and fairly critique the convention to bring it into compliance with the rest of the discipline. In other words, “SOMEONE MAKE A LEGITIMATE RULE AND ENFORCE IT!” Lay down the law instead of allowing this abuse of Order of Operations to tarnish the otherwise marvelous discipline of mathematics!
I understand that part of this comes from linear computer algorithms that take operations as they come. But that is an archaic algorithm now when compared the complexity of language models we have with artificial intelligence (AI). If a language model can figure out the parts of speech of a sentence, then certainly the math algorithms can be overhauled to recognize juxtaposition. The fact that this hasn’t happened yet some 50 years after the advent of the hand-held calculator is astounding to me. We’re basically programming our computers with a primitive, elementary model of mathematics and not accounting for the nuances that arise in Algebra. The order of operations rules should not be different for algebra than they are for arithmetic. Algebra is more or less a theoretical description of arithmetic, so the rules of Algebra should prevail. They should be uniform throughout the field of mathematics.
In my articles above, I go into detail on the influence of juxtaposition in the Arabic number system. For those who haven’t read them yet, here’s a quick summary of the prevalence of juxtaposition and its function in creating a singular value that takes priority in processing.
A fraction is the product of one operand multiplied by the reciprocal of another nonzero operand of the same or different value.[1] The operand below the vinculum (and by definition grouped by the vinculum[2],[3]) should be rationalized, when possible. When looked at this way, then, you have a quantity vertically juxtaposed to a grouped quantity, just like in 8 ÷ 2(2 + 2) you have a quantity (2) horizontally juxtaposed to a grouped quantity. The only difference is the orientation of the juxtaposition.
Yet when dividing by a fraction, we do not pull the numerator/dividend away from the fraction and divide that first, then divide by the denominator/divisor. We treat the fraction as a single quantity, invert the fraction first, then multiply. In other words, the element with a value juxtaposed to another grouped value is addressed first, then the expression is worked (and it’s not insignificant that such a transformation puts multiplication before division, just like it is in PEMDAS). In the same way then, the implied multiplication in 2(2 + 2) MUST be addressed first to be consistent with its cognate function in the former example. The obelus is NOT a grouping symbol, so it is WRONG to treat it as such and give it some nonexistent power to ungroup a juxtaposed quantity.
It is the same way when multiplying or dividing by a mixed number. We do not undo the implied addition of the juxtaposition of a whole number and a fraction but must first convert the mixed number to an improper fraction, then apply the rules for fractions in the previous paragraph.
This is the undeniable reality of how our number system works. I fail to see how anyone can disprove or ignore this reality simply by citing an untested and subjective convention that isolates one of several similar forms and says, “We’re treating it differently because it’s easier than trying to teach it the correct way.” To that I say “Poppycock!”
It’s time to stop capitulating to ambiguity! Mathematics is not worthy of such capitulation. Have some courage and take a stand for what is demonstrably true! Mathematics has hard and fast rules and properties, and the property of juxtaposition should be elevated to the same level as the associative, distributive, and commutative properties. Need a definition? Here it is:
Juxtaposition Property: When two or more values are juxtaposed without any intervening free-standing operators (i.e., operators not included in a grouped value), the juxtaposed values are considered inseparable and must be given priority over extant signed operators in the Order of Operations, regardless of what the implied operators are.
Scott Stocking
My highly informed opinions are my own and are the product of my own research.
[1] To put it another way, a basic division problem using the obelus can be converted to a multiplication problem by inverting (i.e., taking the reciprocal of) the divisor and multiplying by the dividend.
[2] “2 : a straight horizontal mark placed over two or more members of a compound mathematical expression and equivalent to parentheses or brackets about them,” VINCULUM Definition & Meaning – Merriam-Webster, accessed 05/25/25.
[3] “A horizontal line placed above multiple quantities to indicate that they form a unit. vinculum – Wolfram|Alpha, accessed 05/25/25.
A Testimony by Scott Stocking (my views are my own).
Last week, I had the opportunity to see some of Nebraska’s unique monuments and historical sites. I have already written about my experience at Fort Robinson last weekend and how impactful that was to my spiritual well-being. The Sunday after that, I decided I would get a couple places in my National Parks Passport book checked off, essentially since they were relatively close by and no more than a reasonable detour for my trip home.
From Fort Robinson, I headed west on US 20 to Harrison, Nebraska, and from there, I took Nebraska State Highway 29 south toward Mitchell, Nebraska. (I don’t think there’s any hidden meaning in the fact that I had begun my trip west on State Highway 92.) The sign I saw when leaving Harrison said, “No services next 56 miles.” That should let you know just how remote that part of Nebraska is. I saw more tumbleweeds blow across the highway between Harrison and the outskirts of Mitchell than I did residences!
I did find some relief, however, at the Agate Fossil Beds National Monument about halfway between Harrison and Mitchell. The visitor center is about three miles off the highway and was modern and welcoming. I watched a 12-minute movie about the ancient history of the site revealed by the fossil finds, as well as the more recent history of the Native Americans who dwelt in the region. If I had had more time, I would have loved to venture off to one of the dig sites, with the fossil display and the James Cook Gallery (James Cook is the rancher who discovered the fossils).
My next stop was Scotts Bluff National Monument. This is an impressive geological feature along the historic Oregon Trail that served as milestone for travelers along that trail. I had not known that you could drive to the top of the bluff, so I took advantage of that opportunity as well (you can see about 80 miles from the top barring any terrain obstructions). At the top is the placard to Hiram Scott (links to NPS history), who had been abandoned by his employer and died in the vicinity around 1828. The details of his demise are uncertain as we have a few post-mortem accounts of his death up to six years later from various sources that do not agree on every detail. He doesn’t appear to have done anything remarkable or noteworthy other than having been a fur trader, yet his legend was significant enough to ascribe his name to the area, which later became a national monument. In fact, it occurred to me that the sheer face of the bluff is shaped like a giant tombstone.
My final stop at a national monument site on the way home was Chimney Rock National Historic Site (U.S. National Park Service). This has long been an iconic symbol of Nebraska, having the privilege of being the image on the obverse side of the State’s quarter dollar coin. The first thought of most, I think, is that it is the inert remnant of a volcano, given the “chimney” is made of different stuff than the rock it rises from.
According to the NPS Web page, Chimney Rock is the most-referenced geographical feature in pioneer diaries. One quote about it compares it to an ancient pyramid. I bought a T-shirt there that suggests maybe there’s a flying saucer hidden under the structure and the spire is its antenna! I think it would have been a much better site for filming Close Encounters of the Third Kind! While I’m waxing fancifully about conspiracy theories, I might as well speculate that maybe Sasquatch has a secret hideout there as well!
None of these fanciful theories detract from the beauty and majesty of single cone surviving in that region amongst all the giant bluffs and buttes one sees to the west of that location along the historic pioneer trails. For those early pioneers, it was a sign that they were on the right track and had some hope of making it to points further west.
Monumental Lessons
I suppose one could make any number of subjective spiritual connections to these monument sites depending on their life experiences. Many years ago, I wrote about my own “exodus” from Egypt that was Illinois, governed primarily by corrupt Chicago-machine politics with an unemployment rate twice that of Nebraska. If ever there was an argument for a governor of a State to be elected by an electoral college (and Chicago only gets one elector!!) instead of the popular vote, it would be Illinois. In that article (A Tale of Two Photos | Sunday Morning Greek Blog), I wrote about the fiery red sky I’d taken a picture of, with the clouds moving west (usually they would move east).
After the fact, I interpreted that as God calling me back to Nebraska (a Big Red cloud bank moving west, get it?). It was around that time I started hearing the Nebraska fight song in my ears, and it was NOT the ring tone on my flip phone. When I remarried in 2014, my wife and I made annual trips to her family’s cabin in Poudre Canyon. We would fly along I-80 at 75+ mph toward Cheyenne, but I always wished that we’d had a little more time to take the long way around to see Chimney Rock and Scotts Bluff, places I hadn’t seen since my childhood.
Seeing Chimney Rock especially reaffirms the decision I made 15 years ago to follow that “Big Red” cloud west back to Omaha, where I grew up. But I also knew I needed to eventually get to Chimney Rock, that signature geographic wonder in my home State, to have a fuller experience of what it means to be a proud son of Nebraska.
Although Scotts Bluff is named after someone who was just doing his job as a fur trader and had the misfortune to die in the valley near the bluff (the monument to him is on top of the bluff, but the legend says he died somewhere along the North Platte River), I think I can find a more suitable spiritual lesson for myself. The sheer face of the bluff is probably not suitable for rock climbing, that doesn’t mean you can’t get to the top. If I’d had the time and a way to protect myself from rattlesnakes, I might have walked the short trail up to the top. Instead, I took the easy way, a road that winds up to the top traversing through three tunnels along the way.
For me the lesson is simple, and perhaps overly simple enough to be a bit cheesy, but ascending to the top reminds me that I should never stop reaching for new heights in my spiritual journey. One thing I learned on this trip, both at Scotts Bluff and traversing Sow Belly Canyon in NW Nebraska, when you reach new heights, you have a better view of who you are in God’s creation. We humans are “fearfully and wonderfully made,” but to view the vast expanse and beauty of his creation he’s entrusted us with is indeed humbling.
I am grateful that I’m not living in a Matrix-like cocoon (or a Paradise-like underground city) where some sophisticated software engineer has created a simulation for us to live for who-knows-what purpose. It is good to touch and see and smell and feel and hear the wonders of God’s creation all around. No simulation could ever hope to capture the incredible diversity God has granted us in his creation. With views like I witnessed on my swing through western Nebraska, it’s easy to see how the psalmist could write Psalm 148 NIV | Bible Gateway.
Finally, I think I can even draw a spiritual lesson from the Agate Fossil Beds. I’m a young-earth creationist, so I don’t for a moment believe the fossils of ancient animals are 19 million years old. A global flood or a massive caldera explosion (like the one in Yellowstone that’s due!) would have buried those animals rapidly and given the appearance of great age, at least in the modern scientific paradigm.
I wish I could say all my old man behaviors have become extinct, but I know I still struggle from time to time. The good news is, God has not left me alone to deal with behaviors and attitudes that could lead to my untimely extinction. He’s given me life and hope in Jesus for a brighter future and a resurrection from the dead, one that will NOT leave my old bones in the grave, but like Jesus himself experienced, one that is a complete transition from our earthly bodies to our heavenly bodies. And unlike the Visitor Center there at Agate, I do not need a museum display to remind me of my past sins. God has separated them from me as far as the east is from the west.
Agate, Scotts Bluff, and Chimney Rock are all monuments of my faith, and my Fort Robinson trip (Sowbelly Elegy: The Majesty of God in Exegesis | Sunday Morning Greek Blog) reflects, in part, my community of fellowship at this time in my life. They remind me where I’ve come from and what I can look forward to; what I’ve lost and what I’ve gained; who I am and whose I am. I look forward to taking the trip again next year and mixing up the trip home to see more of the beauty of this great State I live in. I trust and pray that you have some special places you can go to have your own “Sabbath Rest” and reconnect with yourself and with God’s son, Jesus, the resurrected one.
I preached this message May 18, 2025, at Mount View Presbyterian Church, Omaha, NE. Fifth Sunday after Easter, Year C.
I find it interesting that in the weeks after Easter, the gospel passages in the Lectionary are revisiting Jesus’s Holy Week events. That probably shouldn’t surprise us with John’s gospel, though, as the last half of his gospel deals with the events of Holy Week. One explanation for this, I think, is that Jesus taught his disciples so much in that last week, and much of it occurred, apparently, immediately after the “Last Supper.” Given what happened in the 24 hours that followed that last supper, I think it’s safe to say that the apostles probably didn’t remember too much of that teaching. It’s a good thing John wrote it down, then! This gives them the opportunity to revisit those precious final moments with Jesus and to review his teachings to see what they missed about his death and resurrection.
Since we’re going back to Holy Week, and especially since today’s passage comes after John’s unique account of the Last Supper, I think it’s worth it to take a look at his account, especially, and add in the details that Matthew, Mark, and Luke provide. At the beginning of John 13, we see that the meal is already in progress, but we don’t get the “ritual” language we’ve become accustomed to from the other three gospels.
There’s no “This is my body” or “This cup is the new covenant in my blood poured out for the forgiveness of sins” in John’s gospel. That’s not to say there’s a contradiction here in the storyline: John focuses on a more radical form of demonstrating the forgiveness that would come from the shed blood of the Messiah. He tells us that the Messiah himself washes the feet of ALL the disciples. When Jesus gets to Peter, we find out a little more about Jesus’s motivation for doing this: “Unless I wash you, you have not part with me.” Jesus turns this act of service into a living, “practical” memorial that his disciples would not soon forget. Not only has he said his blood would bring forgiveness; he touches each one of the disciples, even Judas, who he knows will sell him and out, and Peter the denier, to give them “muscle memory” of forgiveness.
Matthew, Mark, and Luke all say something about the bread and the cup. Matthew and Mark both say simply: “This is my body,” while Luke adds two extra phrases: “This is my body given for you; do this in remembrance of me” (Luke 22:19). Notice that none of the gospel writers ever say, “This is my body broken for you,” although the piece of unleavened bread in this part of the ceremony was the only one formally broken. The church through history almost naturally added in that bit about “broken for you” to parallel what happens to the bread. Note also the references to the cup in the three synoptic Gospel accounts have Jesus saying that the wine is “the new covenant in my blood” or “the blood of my covenant.” Matthew is the only one who connects the blood with the forgiveness of sins.
One thing that Jesus says in all three gospel accounts may get overlooked: “I will not drink from this fruit of the vine from now on until that day when I drink it new with you in my Father’s kingdom.”[1] Jesus here is looking far beyond his own time with this statement: he’s looking ahead to his second coming where we will share in the glorious feast of the Lamb with him in heaven. It’s also worth noting here that Jesus still considers what he’s drinking is “the fruit of the vine” and NOT blood at all.
But this also begs the question: what does Jesus mean when he says, “This IS my body” and “This IS the new covenant in my blood”? I think as Presbyterians we can agree there is not some mystical transubstantiation of the wine into Jesus’s blood. Nor is there a mystical transubstantiation of the bread into the flesh of Christ. But I also don’t think the cup and the bread are merely “symbols” either. I prefer to use the word “signify” to describe the elements because they do have significance for my faith.
In this way, communion is akin to baptism. What does Paul say about baptism in Romans 6:3? “Don’t you know that all of us who were baptized into Christ Jesus were baptized into his death?”[2] Baptism signifies (there’s that word again) that we have come in contact with blood of Christ, which Matthew affirms is for the forgiveness of sins. That is our “initiation” rite, a marker or monument, if you will, that God has done something special in our lives and that we are set apart for something special. The water doesn’t become the blood of Christ when we’re baptized. But in a way that only God knows, the waters of baptism are infused with the power of spiritual cleansing and renewal.
Communion, then, is our regular connection with our baptism, because in communion, as we’ve said, we also encounter the blood of Christ, or what it signifies, in the cup at the communion table. The bread reminds us of the physical suffering Christ endured on the cross. But it also reminds us that we are all part of the body of Christ as well—that’s why we take it together, whether it’s monthly, weekly, or whenever we gather in his name. Out of all the different denominations out there, communion reminds us what we have in common: faith in Christ.
I’ve been studying what the Bible says about communion for quite a long time. It was the topic of one of my early blog posts. In my home church, we take it every Sunday, because that seems to be the practice of the early church in Acts. But my church also typically qualifies it when giving the communion meditation: “If you’re a believer in Christ, we invite you to participate.” There’s no official check for membership or a communicant’s card. Just a simple question to be answered on your honor. Some denominations or branches of mainline denominations require you to be a member of the church. Others may even suggest you’re committing heresy or blasphemy if you take communion in a church where you’re not a member.
The variety in how communion or the eucharist is handled in the modern church concerns me. Communion should be about what Jesus accomplished on the cross, not about your personal affiliation with a particular church. In that early blog article, A Truly Open Communion?, I asked the question this way:
If Jesus calls sinners to himself and eats with them; if Jesus broke bread at the Last Supper with a table full of betrayers and deserters; if Jesus can feed 5,000 men in addition to the women and children with just a few loaves of bread and some fish; why do many churches officially prohibit the Lord’s Table (communion, Eucharist) from those who are not professed Christ-followers, or worse, from those professed Christ-followers who are struggling with sin or divorce or other problems?
Should we really be denying or discouraging those who come to church looking for forgiveness and a connection to the body of Christ the very elements that Jesus uses to signify those things—the bread and the cup? Author John Mark Hicks says this in talking about communion as a “missional table”:
The table is a place where Jesus receives sinners and confronts the righteous; a place where Jesus extends grace to seekers but condemns the self-righteous. Jesus is willing to eat with sinners in order to invite them into the kingdom, but he points out the discontinuity between humanity’s tables…and the table in the kingdom of God.[3]
The implication here is that Jesus is in our midst in a special way, I think, not just because “two or three are gathered in his name,” but because we are doing this “in remembrance” of Jesus. The Old Testament concept of “remembering” is what is key here. In the Old Testament, when the writer says something like, “Then God remembered his promise to Abraham” or “Then God remembered his covenant with Israel,” this not God just calling a set of facts to mind. When God remembers like this, he also acts, and usually in a mighty way.
So when we remember, I believe it is also a call to action on our part, to be empowered by the presence of the Holy Spirit in that moment to make a commitment to action for the days that follow. It’s similar to what Jesus says in the Sermon on the Mount (Matthew 5:23–24): “Therefore, if you are offering your gift at the altar and there remember that your brother or sister has something against you, leave your gift there in front of the altar. First go and be reconciled to them; then come and offer your gift.”[4] By his blood we are forgiven and cleansed to start afresh. By remembering Christ, we are empowered to go out and serve.
As we come to today’s gospel passage again, we find ourselves at the end of the dining part of the Last Supper gathering. Jesus wants this time to be memorable for his disciples, because he tells them this is the last time they’re going to have any meaningful contact with him, at least in his earthly form. John makes a point of saying “When [Judas] was gone,” Jesus began delivering his final instructions, his “action plan” if you will, to give them assurance that they will have the guidance of the Holy Spirit after his death, burial, resurrection, and ascension. All of the elements leading up to his crucifixion have been set in motion, and there’s no turning back now.
That is why Jesus can say, “Now the Son of Man is glorified and God is glorified in him.” He knows what is about to happen. Although Jesus will experience many strong emotions, including betrayal, abandonment, and those associated with excruciating pain, he knows the end result will benefit all mankind for eternity. It’s the day he prepared for but perhaps had hoped would never come, or at least had hoped he would not have to endure alone. He knows the days ahead will be difficult, so he gives them a new command.
“Love one another. As I have loved you, so you must love one another” (John 13:34).
That word “as” carries a huge load with it. It’s not the regular word for “as” in Greek, which is also a two-letter word. The word John uses is a compound word that has the sense of its root words: “love one another according to the way I loved you.” The theme of this new command is found in a few other verses in this part of John, as well as in Luke 6:31: “Treat others according to the way you want to be treated.” “Love” is less about a feeling and more about action. Earlier in John 13, Jesus says, “I have set you an example that you should act toward others according to the way I have acted toward you.” In 15:9, he says, “I have loved you according to the way the Father has loved me,” and then repeats the command from John 13 a few verses later.
Showing this radical, sacrificial, agape kind of love that expects nothing in return is how we show the world we are Jesus followers. It calls us in some cases to reach out beyond our comfort zones and to be hospitable and welcoming to strangers. The author of Hebrews exhorts us in this way in 13:1: “Keep on loving one another as brothers and sisters. 2 Do not forget to show hospitality to strangers, for by so doing some people have shown hospitality to angels without knowing it.”[5] We could all use an angel in our lives from time to time, right? The first chapter of Hebrews (1:14) mentions the function of angels: “Are not all angels ministering spirits sent to serve those who will inherit salvation?”[6] God uses angels, in conjunction with the Holy Spirit, to empower and enable us to show love to others. But I digress just a bit.
God demonstrated his great love for us in Jesus through his life among us, his crucifixion, and his resurrection. I want to bring in the last couple verses of our reading from Psalm 148 this morning, because it is one of the foundational prophecies that show us what the Israelites expected of the Messiah, and Jesus proved faithful to that promise:
Jesus is our horn, the strength that we need to endure each day. Let us continue to hold fast to our Savior so that the world will know him, his salvation, and the power and love of God Almighty. Amen!
[3] Hicks, John Mark. “The Lord’s Supper as Eschatological Table” in Evangelicalism and the Stone-Campbell Movement, Volume 2: Engaging Basic Christian Doctrine. William R. Baker, ed. Abilene: ACU Press, 2006.
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